Action of PGL(2,q) on the Cosets of the Centralizer of an Eliptic Element

Most researchers consider the action of projective general group on the cosets of its maximal subgroups leaving out non-maximal subgroups. In this paper, we consider the action of centralizer of an elliptic element which is a non maximal subgroup . In particular, we determine the subdegrees, rank and properties of the suborbital graphs of the action. We achieve this through the application of the action of a group by conjugation. We have proved that the rank is q and the subdegrees are and .

Compact connected components in relative character varieties of punctured spheres

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

GL(n, q)-analogues of factorization problems in the symmetric group

International audience We consider GLn (Fq)-analogues of certain factorization problems in the symmetric group Sn: ratherthan counting factorizations of the long cycle(1,2, . . . , n) given the number of cycles of each factor, we countfactorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis.Our work generalizes several recent results on factorizations in GLn (Fq) and also uses a character-based approach.We end with an asymptotic application and some questions.

The Sorting Method of the same Pattern Compound Material Based on Basic Element Model of Surface Microstructure

Based on the elliptic element model, the new sorting method on the same pattern compound materials was given in this paper according to the pictures obtained by electronic microscopy systems on the surfaces. Here, the block design combined with variance analysis was introduced to sort the compound materials that contained that the copper-base mingled with graphite, silica and other non-pronounced components. Here, the five kinds of the same pattern compound materials were proposed for sorting, only the graphite distribution was considered and their mixture ratios of components were separately 10%, 15%, 20%, 25%, 30%. At last, through experiment data and statistical analysis, the result of sorting was basically corresponding to the above mixture ratios of components.

Discreteness of subgroups of $\mathrm{SL}(2,\boldsymbol{C})$ containing elliptic elements

The following result is the main result of the paper. Let $G\subset \boldsymbol{SL}(2,\boldsymbol C)$ be non-elementary. If $G$ contains an elliptic element of order at least 3, then $G$ is discrete if and only if each non-elementary subgroup generated by two elliptic elements of $G$ is discrete.

GENERATING SYSTEMS OF SUBGROUPS IN PSL(2,Γn)

It is proved in this paper that for any non-elementary subgroup $G$ of $\mathrm{PSL}(2,\sGa_n)$, which has no elliptic element, to be not strict, there is a minimal generating system of $G$ consisting of loxodromic elements, and that if $G$ is a non-elementary subgroup of $\mathrm{PSL}(2,\sGa_n)$ of which each loxodromic element is hyperbolic, then $G$ is conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$.AMS 2000 Mathematics subject classification: Primary 30F40. Secondary 20H10